Factors, Multiples and Primes - Short Problems


This is part of our collection of Short Problems.

You may also be interested in our longer problems on Factors, Multiples and Primes.

Printable worksheets containing selections of these problems are available here:

Stage 3 ★ Sheet 1 Solutions            Stage 3 ★★ Sheet 1 Solutions
  Sheet 2 Solutions     Sheet 2 Solutions
          Sheet 3 Solutions
Stage 4 ★★★ Sheet 1 Solutions   Stage 3 ★★★ Sheet 1 Solutions

Calculation 2000

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 49 - 2013
What is the value of 2000 + 1999 × 2000?

Flora the Florist

KS 2 & 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 45 - 2009
Flora has roses in three colours. What is the greatest number of identical bunches she can make, using all the flowers?

Last-but-one

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 1 - 2016
What is the last-but-one digit of 99! ?

Almost a Million

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 26 - 2014
Which of the given numbers is divisible by 6?

Adjacent Factors

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 5 - 2011
Two numbers can be placed adjacent if one of them divides the other. Using only $1,...,10$, can you write the longest such list?

Grandma's Cake

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 7 - 2016
What is the smallest number of pieces grandma should cut her cake into to guarentee each grandchild gets the same amount of cake and none is left over.

Prime Order

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 24 - 2006
How many of the rearrangements of the digits 1, 3 and 5 give prime numbers?

End of a Prime

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 25 - 2016
A list is made of every digit that is the units digit of at least one prime number. How many digits appear in the list?

Essential Supplies

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 25 - 2006
Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?

Producing Zeros

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 10 - 2008
If the numbers 1 to 10 are all multiplied together, how many zeros are at the end of the answer?

Stamp Collecting

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 28 - 2016
Last week, Tom and Sophie bought some stamps for their collections. Each stamp Tom bought cost him £1.10, whilst Sophie paid 70p for each of her stamps. Between them they spent exactly £10. How many stamps did they buy in total?

Tricky Customer

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 6 - 2015
Charlie doesn't want his new house number to be divisible by 3 or 5. How many choices of house does he have?

Punky Fish

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 12 - 2008
A male punky fish has 9 stripes and a female punky fish has 8 stripes. I count 86 stripes on the fish in my tank. What is the ratio of male fish to female fish?

Threes and Fours

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 32 - 2016
What is the smallest integer which has every digit a 3 or a 4 and is divisible by both 3 and 4?

Product 100

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 38 - 2010
The product of four different positive integers is 100. What is the sum of these four integers?

Relative Time

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 25 - 2014
Albert Einstein is experimenting with two unusual clocks. At what time do they next agree?

Pairing Up

KS 3 Short Challenge Level:

Challenge Level:1

Weekly Problem 14 - 2008
The numbers 72, 8, 24, 10, 5, 45, 36, 15 are grouped in pairs so that each pair has the same product. Which number is paired with 10?

Find from Factors

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 35 - 2006
A number has exactly eight factors, two of which are 21 and 35. What is the number?

Back of the Queue

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2013
What is the remainder when the number 743589×301647 is divided by 5?

School Dinners

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 1 - 2009
Our school dinners offer the same choice each day, and each day I try a new option. How long will it be before I eat the same meal again?

Sticky Fingers

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 7 - 2011
Ruth wants to puts stickers on the cuboid she has made from little cubes. Will she have any stickers left over?

Coin Collection

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 5 - 2007
When coins are put into piles of six 3 remain and in piles of eight 7 remain. How many remain when they are put into piles of 24?

Factorised Factorial

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

One Short

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 39 - 2014
A whole number less than 100 gives remainders of 2, 3 and 4 when divided by 3, 4 and 5. What is the remainder when it is divided by 7?

Square Sum

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 32 - 2007
One of these numbers is the largest of nine consecutive positive integers whose sum is a perfect square. Which one is it?

Indivisible

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 25 - 2011
Each time a class lines up in different sized groups, a different number of people are left over. How large can the class be?

Powerful Zeros

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 32 - 2006
How many zeros are there at the end of $3^4 \times 4^5 \times 5^6$?

Added Power

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 33 - 2015
How many integers $n$ are there for which $n$ and $n^3+3$ are both prime?

Jenny's Logic

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 52 - 2009
How did Jenny figure out that Tom's cards added to an even number?

Three Primes

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 6 - 2010
Can you find three primes such that their product is exactly five times their sum? Do you think you have found all possibilities?

Missing Digit

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 37 - 2006
What digit must replace the star to make the number a multiple of 11?

17s and 23s

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 22 - 2010
Can you form this 2010-digit number...

Cinema Costs

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 41 - 2009
At a cinema a child's ticket costs £4.20 and an adult's ticket costs £7.70. How much did is cost this group of adults and children to see a film?

Four or Five

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 19 - 2012
The diagram shows a large rectangle composed of 9 smaller rectangles. If each of these rectangles has integer sides, what could the area of the large rectangle be?

Cakes and Buns

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 41 - 2006
Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?

Seven from Nine

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 21 - 2013
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?

Expanding Zeros

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 22 - 2008
The following sequence continues indefinitely... Which of these integers is a multiple of 81?

Multiple Years

KS 3 & 4 Short Challenge Level:

Challenge Level:1

Weekly Problem 18 - 2016
The year 2010 is one in which the sum of the digits is a factor of the year itself. What is the next year that has the same property?

Trailing Zeros

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 13 - 2009
How many zeros does 50! have at the end?

Divisible Palindrome

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 5 - 2014
What is the sum of the digits of the largest 4-digit palindromic number which is divisible by 15?

Integer and Integer

KS 3 & 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 39 - 2012
For how many values of $n$ are both $n$ and $\frac{n+3}{n−1}$ integers?

Factor List

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 48 - 2010
Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

Supercomputer

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 28 - 2014
What is the units digit of the given expression?

Square Product

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 10 - 2011
Will this product give a perfect square?

HCF Expression

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Weekly Problem 30 - 2010
Find out which two distinct primes less than $7$ will give the largest highest common factor of these two expressions.

Square LCM

KS 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 7 - 2010
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?

Primes and Six

KS 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Weekly Problem 1 - 2015
If $p$ and $q$ are prime numbers greater than $3$ and $q=p+2$, prove that $pq+1$ is divisible by $36$.